Physics 203 – College Physics I Department of Physics – The Citadel Physics 203 College Physics I Fall 2012 S. A. Yost Chapter 3 Motion in 2 Dimensions – Part 1

Physics 203 – College Physics I Department of Physics – The Citadel Today’s Topics Vectors We will introduce the concept of vectors, which have many applications throughout physics, and are the most important new mathematical concept used in the course.

Physics 203 – College Physics I Department of Physics – The Citadel Thursday’s Assignment Read Ch. 3, except section 8. A problem set on HW3 on Ch. 3 will be due next Tuesday. The first exam is now scheduled for Thursday, Sept. 20. The calendar in the syllabus posted on CitLearn has been updated. You do not need to memorize equations: the essential ones will be provided for the exam.

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 2 Which of the equations gives the correct relation between the vectors in the figure? A. A + B + C = 0 B. A = B + C C. B = A + C D. C = A + B E. None of these B C A → → → → → →

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 1 Which of the following is a vector? A. Mass B. Temperature C. Distance D. Displacement E. Speed

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 3 Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B? A. Always B. When A and B point in opposite directions. C. Never D. When A and B are parallel. E. When A and B are perpendicular. → → → → →

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 4 Vector A has a magnitude of 10 and a direction angle θ = 60 o measured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A? A. – 20, 60 o B. 20, 240 o C. 20, – 30 o D. – 20, 240 o E. – 20, – 30 o → → x y A → θ = 60 o

Physics 203 – College Physics I Department of Physics – The Citadel Vectors and Scalars Scalars are quantities described entirely by a number, with no need to specify a direction – the temperature, for example. Vectors require both a magnitude and direction to be fully specified. Describing motion in 2 or more dimensions requires vectors. Also forces, which must act in some direction, are described by vectors.

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 1 Which of the following is a vector? A. Mass B. Temperature C. Distance D. Displacement E. Speed

Physics 203 – College Physics I Department of Physics – The Citadel The position of a point B relative to a point A is given by a displacement vector D pointing from A to B. This vector tells you how to get from point A to point B. Displacement Vectors D → → A B

Physics 203 – College Physics I Department of Physics – The Citadel Cartesian coordinates are used to label points in a plane. The lengths of a vector along the two axes are called its Cartesian components. D x = 2, D y = 5. Cartesian Components x y 0 D → DxDx DyDy

Physics 203 – College Physics I Department of Physics – The Citadel A vector can also be specified by giving its magnitude and direction. The magnitude is the length of the vector: D = |D|. The direction can be given by an angle relative to an axis. The angle in polar coordinates is measured counterclockwise from the x axis. Polar Coordinates x y 0 D → → θ

Physics 203 – College Physics I Department of Physics – The Citadel Mathematical Review: Right Triangle The sides of a right triangle satisfy the Pythagorean Theorem: a 2 + b 2 = c 2 c b a

Physics 203 – College Physics I Department of Physics – The Citadel Mathematical Review: Trigonometry The ratios of sides of a right triangle define the trigonometric functions. sin θ = b/c cos θ = a/c tan θ = b/a csc θ = c/b sec θ = c/a cot θ = a/b Inverses: θ = asin (b/c) = acos(a/c) = atan(b/a) c b a θ

Physics 203 – College Physics I Department of Physics – The Citadel Polar Coordinates x y 0 D → → θ Find the magnitude and direction of D. D x = 2, D y = 5 D = √ D x 2 + D y 2 = √29 = 5.4 tan θ = 5/2 = 2.5 θ = tan  1 (2.5) = 68 o

Physics 203 – College Physics I Department of Physics – The Citadel Geometrically, two vectors are added by following one to the end, then following the second from that point, and finding the net displacement. Components: = + Vector Addition A → B → C → C → B → A → C x = A x + B x C y = A y + B y

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 2 Which of the equations gives the correct relation between the vectors in the figure? A. A + B + C = 0 B. A = B + C C. B = A + C D. C = A + B E. None of these C A → B → → → → →

Physics 203 – College Physics I Department of Physics – The Citadel Vectors Two vectors, A and B, of length 5 and 3 respectively, lie in a plane, but the directions are unspecified. What is the maximum magnitude of A + B? |A+B| = 8 What is the minimum magnitude of A + B? |A+ B|=2 B → A → C → C → B → A → →

Physics 203 – College Physics I Department of Physics – The Citadel Scalar Multiple Vectors can be multiplied by scalars (numbers). Multiplying by a positive number changes the length, not the direction: Multiplying by a negative number also changes the direction by 180 o : A → 2A → A → – A →

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 4 Vector A has a magnitude of 10 and a direction angle θ = 60 o measured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A? A. – 20, 60 o B. 20, 240 o C. 20, – 30 o D. – 20, 240 o E. – 20, – 30 o → → x y A → θ = 60 o

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 4 Vector A has a magnitude of 10 and a direction angle θ = 60 o measured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A? A. – 20, 60 o B. 20, 240 o C. 20, – 30 o D. – 20, 240 o E. – 20, – 30 o → → x y A → θ = 60 o – 2A → θ = 240 o 20 10

Physics 203 – College Physics I Department of Physics – The Citadel Vector Difference The vector difference A – B can be formed by adding the vector – B to the vector A. A – B can be interpreted as the displacement that takes you from B to A. A → B → – B → A – B →→ → → →→ → → → →

Physics 203 – College Physics I Department of Physics – The Citadel Quiz: Question 3 Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B? A. Always B. When A and B point in opposite directions. C. Never D. When A and B are parallel. E. When A and B are perpendicular. → → → → → A → B → C →